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Notes on random walks in the Cauchy domain of attraction

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Abstract

The goal of these notes is to fill some gaps in the literature about random walks in the Cauchy domain of attraction, which has often been left aside because of its additional technical difficulties. We prove here several results in that case: a Fuk–Nagaev inequality and a local version of it; a large deviation theorem; two types of local large deviation theorems. We also derive two important applications of these results: a sharp estimate of the tail of the first ladder epochs, and renewal theorems. Most of our techniques carry through to the case of random walks in the domain of attraction of an \(\alpha \)-stable law with \(\alpha \in (0,2)\), so we also present results in that case—some of them are improvement of the existing literature.

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Acknowledgements

I am most grateful to Vitali Wachtel for his comments and his suggestions for the improvement of Theorem 3.2, and also to I. Kortchemski and L. Richier for attracting my attention to some subtleties of Theorem 3.4 (and to their article [19]). I thank the referees for their remarks and suggestions, in particular for pointing out references (and an elementary proof) for Theorem 3.7. I am grateful to Thomas Duquesne and Francesco Caravenna for the many discussions we had on this (and related) topic(s), and I also thank Cyril Marzouk for telling me about the reference [5].

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  1. LPSM, Sorbonne Université, Campus Pierre et Marie Curie, Case 158, 4 Place Jussieu, 75252, Paris Cedex 5, France

    Quentin Berger

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  1. Quentin Berger
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Berger, Q. Notes on random walks in the Cauchy domain of attraction. Probab. Theory Relat. Fields 175, 1–44 (2019). https://doi.org/10.1007/s00440-018-0887-0

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